Explanation Given: Key concepts applied: We need to draw triangles from the figure and label the distances as shown in the figure below: By repeated application of the Pythagorean Theorem, We find ( d + R ) 2 = k 2 + R 2 → 1 s t E q u a t i o n (Applying Pythagorean Theorem to Bigger Triangle) k 2 = ( d + h ) 2 + x 2 → 2 n d E q u a t i o n (Applying Pythagorean Theorem to Medium Triangle) R 2 = x 2 + ( R − h ) 2 → 3 r d E q u a t i o n (Applying Pythagorean Theorem to Smaller Triangle) Substituting the 2 nd and 3 rd equation in 1 st equation we get… ( d + R ) 2 = ( d + h ) 2 + x 2 + x 2 + ( R − h ) 2 → 4 s t E q u a t i o n Formula used: Pythagorean Theorem: a 2 + b 2 = c 2 (where a and b are legs of Right Triangle and c is Hypotenuse) Algebraic Identities: ( m + n ) 2 = m 2 + n 2 + 2 m n ( m − n ) 2 = m 2 + n 2 − 2 m n Calculation: Solving for h yields… ( d + R ) 2 = ( d + h ) 2 + x 2 + x 2 + ( R − h ) 2 d 2 + R 2 + 2 d R = d 2 + 2 d h + h 2 + 2 x 2 + R 2 + h 2 − 2 R h G e t t i n g a l l t h e t e r m s i n o n e s i d e o f t h e e q u a t i o n , w e g e t d 2 + R 2 + 2 d R − ( d 2 + 2 d h + h 2 + 2 x 2 + R 2 + h 2 − 2 R h ) = 0 D i s t r i b u t i n g t h e n e g a t i v e d 2 + R 2 + 2 d R − d 2 − 2 d h − h 2 − 2 x 2 − R 2 − h 2 + 2 R h = 0 C o m b i n i n g l i k e t e r m s d 2 + R 2 + 2 d R − d

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Chapter 9.2, Problem 62E

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Using applying the Pythagorean Theorem to the three right triangles in given figure that $h = \frac{d R}{d + R}$

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Chapter 9.2, Problem 61E

Chapter 9.2, Problem 63E

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Using applying the Pythagorean Theorem to the three right triangles in given figure that h = d R d + R

Solution Summary: The author illustrates how the Pythagorean Theorem is applied to the three right triangles in the figure.

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Using applying the Pythagorean Theorem to the three right triangles in given figure that $h = \frac{d R}{d + R}$

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Using applying the Pythagorean Theorem to the three right triangles in given figure that h = d R d + R

Solution Summary: The author illustrates how the Pythagorean Theorem is applied to the three right triangles in the figure.

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Chapter 9 Solutions

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Using applying the Pythagorean Theorem to the three right triangles in given figure that $h = \frac{d R}{d + R}$